If the USA is anything like the UK, Indian illegals come over to the UK and are exploited by their own to work for cut price jobs in restaurants, factories and even fruit pickers. So if legitimate businesses do not employ them, which is very likely due to a lack of paperwork, there are plenty of illegal employers.
It's a little different in this case since most of the illegal employers are actually highly influential agrobusinesses who have either perfected loophole-jumping to an exact science or are simply too rich for anyone to bother prosecuting them.
But your basic point is correct. They're coming because the jobs are being offered to them, implicitly or even explicitly. If American businesses stopped hiring them, they'd stop coming.
The Bureau of Immigration could keep copies of all work permits/IDs of those permitted to work on an online data base. The Employer will then access this with the ID Number as a reference. No record no employment. The records would be accessible to immigration control when checking passports on entry.
And I would oppose this--rather than keeping illegals out this will mean forged documents + identity theft rather than just forged documents. We need to do a better job with identity theft before we go this route.
We ALREADY HAVE a problem with identity theft. Actually, it seems like keeping track of work permits and visas in a centralized database would make it that much easier to catch the thieves.